Technorati

Search this blog

Saturday, August 20, 2011

The times table and the number line

“Number sense” is a short-hand for our ability to quickly understand, approximate, and manipulate numerical quantities.

[snip]

My hypothesis is that number sense qualifies as a biologically determined category of knowledge. I propose that the foundations of arithmetic lie in our ability to mentally represent and manipulate numerosities on a mental “number line”, an analogical representation of number; and that this representation has a long evolutionary history and a specific cerebral substrate.

[snip]

The hypothesis developed in TNS [The Number Sense] is that all children are born with a quantity representation which provides the core meaning of numerical quantity. Exposure to a given language, culture, and mathematical education leads to the acquisition of additional domains of competence such as a lexicon of number words, a set of digits for written notation, procedures for multi-digit calculation, and so on. Not only must these abilities be internalized and routinized; but above all, they need to be coordinated with existing conceptual representations of arithmetic. The constant dialogue, within the child’s own brain, between linguistic, symbolic, and analogical codes for numbers eventually leads, in numerate adults, to an integrated set of circuits that function with an appearance of non-modularity. Before such a flexible integration is achieved, however, the hypothesis of a modularity and lack of coordination of number representations can explain many of the systematic errors or difficulties that children encounter in the acquisition of arithmetic.

[snip]

Here I shall only discuss one example, the memorization of the multiplication table. Why is it so difficult to learn the small number of single-digit multiplication facts? Leaving out multiplications by 0 and by 1, which can be solved by a general rule, and taking into account the commutativity of multiplication, there are only 36 facts such 3x9=27 that need to be learned. Yet behavioral evidence indicates that even adults still make over 10% errors and respond in more that one second to this highly overtrained task.

What I think is happening is that our intuition of quantity is of very little use when trying to learn multiplication. Approximate addition can implemented by juxtaposition of magnitudes on the internal number line, but no such algorithm seems to be readily available for multiplication. The organization of our mental number line may therefore make it difficult, if not impossible, for us to acquire a systematic intuition of quantities that enter in a multiplicative relation. (This hypothesis is supported by the fact that patients can have severe deficits of multiplication while leaving number sense relatively intact; in particular, patient NAU (Dehaene & Cohen, 1991), who could still understand approximate quantities despite aphasia and acalculia, was totally unable to approximate multiplication problems). In order to memorize multiplication facts, we therefore have to resort to other strategies based on nonquantitative number representations. The strategy that cultures throughout the world have converged on is to acquire multiplication facts by rote verbal learning. That is, each multiplication fact is recited and remembered as a rote phrase, a specific sequence of words in the language of teaching. This is still a difficult task, however, because the 36 facts to be learned all involve the same number words in slightly different orders, with misleading rhymes and partial overlap. Error analyses indeed indicate that interference in memory is the most frequent cause of multiplication error. When we err, say, on 7x8, we do not produce 55 or 57, which would be close matches, but we typically say 63, which is the correct multiplication result of the wrong operation, 7x9 (Ashcraft, 1992; Campbell & Oliphant, 1992).

It can be argued that our memory never evolved to acquire a lot of tightly inter-related and overlapping facts, as is typical of the multiplication table. Our long-term semantic memory is associative and content-adressable: when cued with a specific episode, we readily retrieve memories of related contents based on the semantic similarity. In particular, we generalize approximate additions based on numerical proximity. Hence, we can readily reject 34+47=268 as false, even though we have never been exposed to this particular fact, because our representation of quantity immediately allows us to recognize that the proposed quantity, 268, is too distant from the operands of the addition (Ashcraft & Battaglia, 1978; Dehaene et al., 1999). In the case of exact multiplication, however, the organization of memory by proximity is detrimental to performance. It would be desirable to keep each multiplication fact separate from the others ; yet our memory is designed so that, when we think of 6x7, we co-activate 6x8 and 5x7. In summary, our cerebral organization can explain both why exact multiplication facts are so confusing and difficult to learn, and why approximation and understanding of quantities are highly intuitive operations.

This is pretty much Wayne Wickelgren's explanation as to why it's difficult to memorize the times tables.

My own systematic error in multiplication, on the other hand, directly contradicts Dehaene's observations: until 5 years ago, I had spent my entire adult life beileiving that 7x6=43. (As I recall, Vlorbik was one of the people who alerted me to the fact that 7x6=42, back on the old site.)

If I were a normal person I would have spent my entire adult live believing that 7x6 was 49 or possibly 35.

14 comments:

Anonymous
said...

And the classic Ernst Kummer story:

"Ernst Eduard Kummer (1810-1893), a German algebraist, was rather poor at arithmetic. Whenever he had occasion to do simple arithmetic in class, he would get his students to help him. Once he had to find 7 x 9. 'Seven times nine,' he began, 'Seven times nine is er -- ah --- ah -- seven times nine is. . . .' 'Sixty-one,' a student suggested. Kummer wrote 61 on the board. 'Sir,' said another student, 'it should be sixty-nine.' 'Come,come, gentlemen, it can't be both,' Kummer exclaimed. 'It must be one or the other.

My hypothesis is that shoe tying qualifies as a biologically determined category ofknowledge. My son always tied his shoes once when we bought them, and never tied them again. We struggled to get him to untie his shoes before taking them off, but it was hopeless. We complained that he was ruining his shoes, but he always outgrew them before real damage showed up. When he did tie his shoes, it always took him longer than 7 seconds per shoe. He can play the piano and he can do Jacob Ladder in 3 seconds, so that must imply that there is something genetic about shoe tying.

@SteveHMy son couldn't tie shoes either; he just made a big knot and slipped them on and off. When he was around six or seven, though, we explained to him the topology of square knots, and why the ends of the laces were doubled back for the second part of the knot.Then he could do it, but decided it was too much trouble (and now, nearly 40, wears sandals with Velcro most of the year).--Linda Seebach

Oh, and what's the explanation for why children are so poor at approximation or understanding of quantities?

My thought exactly, especially having in mind my recent student (8 years old, normal intelligence) who could not identify simple quantities without counting each time? For example, if shown a picture of 1 thing, a picture of 4 things and a picture of 30 things (all the same item), he could not tell you which was the picture of 4 without manually counting. He could not even look at the picture of 1 and stay, that's not 4. He had to count the 1 to arrive at that conclusion.

If "all children are born with a quantity representation which provides the core meaning of numerical quantity", what can be the explanation?

In regards to Catherine's mistake of 6x7=43, if one of my kids gave that answer, I'd ask if it made sense for a multiple of an even number to be odd.

Sometimes it feels like my primary task in homeschooling math is to get my kids to actually double-check their answers to see if they make sense. Here's an example from last week.

"Mrs. Ward bought 840 eggs. She sold them in trays of 12 eggs each. How much money did she receive if the selling price per tray was $3?"

My DD divided 84 not 840 by 12 and wound up with an answer that was off by a factor of ten. If she'd stopped to think for a moment, she'd have realized that the answer should be somewhere in the ballpark of $240 (800/10 * 3).

I have a BS in math and a PhD in computer science, and I still don't know all of my multiplication factoids. I think it is far more important to be able to visualize multiplication, so you know when to use it to solve a problem. I know way too many college students who don't have a firm grasp of the concept of multiplication, though I suspect they could spout their times table factoids with ease.

Back when kids had to tie shoes because there were no other options, they did learn, and much earlier than today. I cannot remember a single kid in my first grade class who could not tie his or her shoes.

I met a classmate of my son's who had the entire multiplication table memorized at age 6, in first grade. He was happy to whip them all off. He had an older sibling that was using recitation to memorize and had overhead her enough to get it down. Fantatic auditory memory. This enabled the lad to play many games on the school server that didn't require actually understanding what multiplication was.

In later years, he had trouble. Because he memorized before he understood, it was difficult to discipline himself to think through the why explanations and internalize the associative, commutative, and distributive properties in third and forth grade, rather than jump to the 'answer'. He did not make it 8th grade algebra.

In regards to Catherine's mistake of 6x7=43, if one of my kids gave that answer, I'd ask if it made sense for a multiple of an even number to be odd.

Yes, and let me tell you I was darned embarrassed when I realized I'd spent my entire adult life without realizing that.

I wish to heck our schools would just go ahead and adopt Singapore Math universally. I was reading the study about elementary school teachers being math-phobic -- well, how likely is it that a math-phobic grade school teacher is going to alert her students to the fact that multiples of even numbers are always even?

Huh -- I realize this is a very old post. (I was specifically looking for it to pass along to a friend) -- but this time when I read it, it was immediately obvious to me why you made the 6x7=43 error -- you were conflating the 6x7 fact with 6+7. I don't know why I didn't see that the first time through.